Optica Applicata, Vol. XL, No. 1, 2010 Optical vortex sign determination using self-interference methods PIOTR KURZYNOWSKI, MONIKA BORWIŃSKA, JAN MASAJADA Institute of Physics, Wrocław University of Technology, Wybrzeże Wyspiańskiego 27, 50-370 Wrocław, Poland We have proposed a simple method for determining the sign of optical vortex seeded in optical beam. Our method can be applied to any single optical vortex, also the one with topological charge magnitude higher than 1, as well as to the whole vortex lattice. The proposed method has been verified experimentally for all the cases. Keywords: optical vortex, interference, birefringence. 1. Introduction Optical vortices are singular lines in a phase distribution of a light field. Usually they are observed in planar cross section (on the screen) as dark points with undefined phase (vortex points) [1–3]. The wavefront takes characteristic helical form in the vicinity of the vortex line (Fig. 1) [1–4]. The twisted helical wavefront results a non-zero angular momentum carried by the beam with the optical vortex [3]. The wavefront’s helical geometry allows for vortex classification due to the helice handedness. We say that optical vortex may have ab Fig. 1. Left (a) and right (b) oriented helical wavefront. The light wave phase is undetermined along the axis of the helice. 166 P. KURZYNOWSKI, M. BORWIŃSKA, J. MASAJADA Fig. 2. Generally the vortex lines in a complex scalar field may have complicated geometry. Here, the line intersects the plane Σ twice. The phase circulation determined in plane Σ in the neighborhood of both intersection points circulates in opposite directions. However, the phase circulations, while looking along vectors n, are constant for the entire line. positive or negative topological charge. The question of topological charge is more complicated than it is usually referred to in literature. Short discussion on this matter may be found in [5]. Generally, singular lines may have complicated geometry [6–8]. In Fig. 2 versors n(s) are unit vectors distributed along the singular line. The choice of the vectors n sense is arbitrary, however, the whole vector field along singular lines must be continuous. The phase circulation splits vortex lines into two classes: left and right handed. As it was mentioned in many previous papers, the fields containing such singular lines are studied in a given plane, which often represents the plane of detection. In such a plane, vortices are classified according to their phase circulation having either positive or negative topological charge. Figure 2 shows that along the single vortex line the positive and negative vortices can be observed while intersecting with the plane Σ. Here the single well defined singular line can be observed as carrying both the positive and negative topological charges. For the monochromatic waves propagating in a well defined direction the topological charge can be identified without ambiguity. For example, such a situation happens in the case of waves generated by a paraxial superposition of laser beams [9, 10]. The results presented in our paper are valid for such cases. Optical vortices have found a number of applications: they play an important role in optical manipulators [11], they can be applied as phase markers in classical and speckle interferometry [10, 12–14], or they can be used for determining the light polarization state [15–17]. They are also used as light suppressing elements in fluorescence microscopy [18]. Suppressing the light of the star using the vortex filter possibly will enable direct observation of the terrestrial planet orbiting around the stars [19, 20]. For most of these applications, the exact determination of the optical vortex sign is not necessary and it is enough to classify observed vortices into two classes: positive and negative. However, for some applications, determining the sign of optical vortices [10, 12–17] is important. For example, knowing the sign distribution Optical vortex sign determination using self-interference methods 167 a b c d Fig. 3. The interference of paraxial beam carrying optical vortex with unit topological charge and inclined plane wave. in vortex lattice generated in vortex interferometer allows for continuous phase determination (without phase unwrapping procedures). In the literature there are several papers devoted to the problem of optical vortex sign determination. In papers [21, 22] the simple interfering methods for determining the sign of a single optical vortex is presented. These methods requires precise adjustment of the interferometric system, otherwise the results are ambiguous. Two methods based on the same concept (Fig. 3) were proposed for determining the vortex sign in the vortex lattice created in optical vortex interferometer [23, 24]. In the present paper the same basic idea is used. Figure 4 explains it in a more detailed way (a bit different figure is shown in paper [24]). The method based on the Young’s diffraction experiment is reported in [22, 25, 26]. The method, in which a planar waveguide is used, is reported in [22, 27]. The method using shearing interferometry was reported in [28, 29]. Interferometric technique combined with polarymetry was proposed in paper [30]. This method gives both vortex charge and vortex position but requires the reference beam, so the optical system is more complicated than in case of shearing methods. This paper reports on a new method for vortex sign determination. The method is based on the interference resulting from splitting incident wave by Wollaston prism or birefringent plate. Depending on the splitting element, the splitting could be angular or lateral. The measurement system is simple and stable – no adjustment and no reference beam is necessary. It can be used for measuring the single vortex sign (also 168 P. KURZYNOWSKI, M. BORWIŃSKA, J. MASAJADA abc d Fig. 4. A negative optical vortex interferes with inclined plane waves (a and b). The phase increment direction is indicated by arrows. In the case (a) the vortex phase increases in the same direction as the plane wave phase in lower part of the figure (below singular point). In the case (b) the same happens in upper part of the figure (above singular point). We may find a point of maximum/minimum intensity below singular point (a) or above singular point (b). The points of maximum/minimum intensity are plotted by dark/light circles. Below/above the singular point phases of both waves increases in the same direction. That means that the dark points (light points) will split to the left and right forming a fork-like fringe. In the opposite part (above/below) the phase increases in opposite directions, so fringes cannot split. Although the vortex sign in both cases (a) and (b) is the same the fork fringe are in opposite directions. Figures c and d show the same situation for vortex with positive charge. with topological charge higher than 1) as well as for determining the vortex charges in the whole vortex lattice as produced, for example, by vortex interferometer. The results are free of ambiguity. 2. Fringe interference – angular sharing setup The concept of our measurement system originates from works on one beam vortex interferometer [15]. The continuation of these works has led us to the idea of a simple system for measuring the light polarization state [16, 17] as well as to the idea of a system for measuring the vortex sign, which is a subject of this paper. Our optical system consists of a Wollaston prism working between the polarizer and the analyzer (Fig. 5). The measured light beam enters the system through the polarizer. The Wollaston prism splits the input wave into two: an ordinary and extraordinary one, so the optical field can be considered as two incident wave copies being inclined at small angles. The angle depends on the Wollaston prism angle and the birefringence of the prism’s material. These two beams interfere with each other after the analyzer, thus the resulting intensity distribution contains two forks-like fringes as it is shown in Fig. 6. Both beams are displaced at the observation plane such that the dark point of one beam meets the bright area of the second one. Consider the small neighborhood of the singular point belonging (say) to the left beam at observation plane. The phase of the bright part of the right beam covering this neighborhood will have strong phase Optical vortex sign determination using self-interference methods 169 Fig. 5. The measurement setup for optical vortex sign identification: angular sharing case. P – polarizer with the azimuth angle 0°, A – analyzer with the azimuth angle 45°, W – Wollaston prism with the azimuth angle 90°. a b c d Fig. 6. The interference calculated numerically for positive (a) and negative (b) optical vortex. An optical vortex (positive (c), negative (d)) was introduced into laser beam after reflection from a spatial light modulator and the beam passed through the measurement system shown in Fig. 5. gradient due to the inclination angle. As a result, in this neighborhood, this small part of the second beam can be treated as a plane wave so one can refer to Fig. 4 when analyzing the experiment. In Figures 3 and 4 one can see that the two copies of the vortex interfere with the non-vortex part of the other beam at opposite inclination. That is why the observed two forks are in opposite orientation. Changing the sign of the vortex seeded into incident beam changes orientation of the forks fringes determining at the same way uniquely the topological charge of the optical vortex. Figure 7 shows the same situation with vortex topological charge of magnitude ±3. 170 P. KURZYNOWSKI, M. BORWIŃSKA, J. MASAJADA a b Fig. 7. The same experiment as in Figs. 6c and 6d for the optical vortex with charge magnitude of ±3.